Chern character, loop Spaces and derived algebraic geometry

Chern Character, Loop Spaces and Derived

Algebraic Geometry

Bertrand To¨en and Gabriele Vezzosi

Abstract In this note we present a work in progress whose main purpose is to

estab-lish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of O-modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme, which is a derived scheme whose theory of functions is closely related to cyclic homology of. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere (see TV1 and TV2).

1 Motivations and Objectives

The purpose of this short note is to present, in a rather informal style, a construction of a Chern character for certain sheaves of categories rather than sheaves of modules (like vector bundles or coherent sheaves). This is part of a more ambitious project to develop a general theory of what we call categorical sheaves, in the context of algebraic geometry but also in topology, which is supposed to be a categorification of the theory of sheaf of modules. Our original motivations for starting such a project come from elliptic cohomology, which we now explain briefly.

B. To¨en (

B

)

Institut de Math´ematiques de Toulouse UMR CNRS 5219, Universit´e Paul Sabatier, France; G. Vezzosi, Dipartimento di Matematica Applicata, Universit`a di Firenze, Italy

e-mail:bertrand.toen@math.univ-toulouse.fr

N.A. Baas et al. (eds.), Algebraic Topology, Abel Symposia 4,

DOI 10.1007/978-3-642-01200-6 11, c Springer-Verlag Berlin Heidelberg 2009

1.1 From Elliptic Cohomology to Categorical Sheaves

To any (complex oriented) generalized cohomology theory (defined on topolog-ical spaces) is associated an integer called its chromatic level, which by definition is the height of the corresponding formal group. The typical generalized coho-mology theory of height zero is singular cohocoho-mology and is represented by the Eleinberg–MacLane spectrum Z. The typical generalized cohomology theory of height 1 is complex K-theory which is represented by the spectrum  Z. A typical cohomology theory of height 2 is represented by an elliptic spectrum and is called elliptic cohomology. These elliptic cohomologies can be combined altogether into a spectrumtmfof topological modular forms (we recommend the excellent survey [Lu1] on the subject). The cohomology theoriesZ and  Z are rather well understood, in the sense that for a finite CW complex is possible to describe the groupsZand  Zeasily in terms of the topology of . Indeed,Z  



Zis the group of continuous functions  Z. In the same way,  Z

* 

is the Grothendieck group of com-plex vector bundles on. As far as we know, it is an open question to describe the grouptmf MM



, or the groupsfor some elliptic spectrum, in similar geometric terms, e.g., as the Grothendieck group of some kind of geo-metric objects over (for some recent works in this direction, see [Ba-Du-Ro] and [St-Te]).

It has already been observed by several authors that the chromatic level of the cohomology theoriesZ and  Z coincide with a certain categorical level. More precisely, Z is the set of continuous functions   Z. In this descriptionZ is a discrete topological space, or equivalently a set, or equivalently a-category. In the same way, classes in  Zcan be represented by finite dimensional complex vector bundles on. A finite dimensional complex vector bundle on is a continuous family of finite dimensional complex vector spaces, or equivalently a continuous map  6L, where6L is the 1-category of finite dimensional complex vector spaces. Such an interpretation of vector bundles can be made rigorous if6L is considered as a topological stack. It is natural to expect thattmfis related in one way or another to -categories, and that classes in tmfshould be represented by certain continuous morphisms  6L, where now6L is a-category (or rather a topological-stack). The nota-tion 6L suggests here that6L is a categorification of6L, which is itself a categorification ofZ (or rather of C). If we follow this idea further the typical generalized cohomology theory of chromatic level $ should itself be related to $-categories in the sense that classes in  should be represented by continuous maps  $6L, where $6L is now a certain topo-logical$-stack, which is supposed to be an$-categorification of the$-stack $6L.

This purely formal observation relating the chromatic level to some, yet unde-fined, categorical level is in fact supported by at least two recent results. On the one hand, Rognes stated the so-called red shift conjecture, which from an intuitive point

of view stipulates that if a commutative ring spectrum is of chromatic level$ then its -theory spectrum is of chromatic level$(see [Au-Ro]). Some explicit computations of   Z proves a major case of this conjecture for $ (see [Ba-Du-Ro]). Moreover,   Zcan be seen to be the -theory spectrum of the-category of complex-vector spaces (in the sense of Kapranov– Voevodsky). This clearly shows the existence of an interesting relation between elliptic cohomology and the notion of-vector bundles (parametrized version of the notion-vector spaces), even though the precise relation remains unclear at the moment. On the other hand, the fact that topological -theory is obtained as the Grothendieck group of vector bundles implies the existence of equivariant -theory by using equivariant vector bundles. It is important to notice here that the spectrum  Z alone is not enough to reconstruct equivariant -theory and that the fact that complex -theory is obtained from a categorical construction is used in an essential way to define equivariant -theory. Recently Lurie constructed not only equivariant versions but also-equivariant versions of elliptic cohomology (see [Lu1, Sect. 5.4]). This means that not only an action of a group can be incorpo-rated in the definition of elliptic cohomology, but also an action of a-group (i.e., of a categorical group). Now, a-group can not act in a very interesting way on an object in a-category, as this action would simply be induced by an action of the group



. However, a-group can definitely act in an interesting manner on an object in a-category, since automorphisms of a given object naturally form a-group. The existence of-equivariant version of elliptic cohomology therefore suggests again a close relation between elliptic cohomology and-categories.

Perhaps it is also not surprising that our original motivation of understanding which are the geometric objects classified by elliptic cohomology, will lead us below to loop spaces (actually a derived version of them, better suited for alge-braic geometry: see below). In fact, the chromatic picture of stable homotopy theory, together with the conjectural higher categorical picture recalled above, has a loop space ladder conjectural picture as well. A chromatic type$cohomology the-ory on a space  should be essentially the same thing as a possibly equivariant chromatic type$cohomology theory on the free loop space, and the same is expected to be true for the geometric objects classified by these cohomology theo-ries (see e.g., [An-Mo, p. 1]). This is already morally true for$ , according to Witten intuition of the geometry of the Dirac operator on the loop space ([Wi]): the elliptic cohomology on is essentially the complex



-equivariant -theory of. Actually, this last statement is not literally correct in general, but requires to stick to elliptic cohomology nearand to the “small” loop space, as observed in [Lu1, Rmk. 5.10].1

1_{A possibly interesting point here is that our derived loop spaces have a closer relation to small}

1.2 Towards a Theory of Categorical Sheaves in Algebraic

Geometry

The conclusion of the observation above is that there should exist an interesting notion of categorical sheaves, which are sheaves of categories rather than sheaves of vector spaces, useful for a geometric description of objects underlying elliptic cohomology. In this work we have been interested in this notion independently of elliptic cohomology and in the context of algebraic geometry rather than topology. Although our final motivations is to understand better elliptic cohomology, we have found the theory of categorical sheaves in algebraic geometry interesting in its own right and think that it deserves a fully independent development.

To be more precise, and to fix ideas, a categorical sheaf theory is required to satisfy the following conditions.

 For any scheme there exists a -category=7, of categorical sheaves on . The-category=7 is expected to be a symmetric monoidal  -category. Moreover, we want=7to be a categorification of the category A-of sheaves ofO

--modules on

, in the sense that there is a natural equivalence between A-and the category of endomorphisms of the unit object in=7.

 The -category =7 comes equipped with monoidal sub--categories =7qcoh,=7coh, and =7parf, which are categorifications of the categories [=A , =A , and 6L, of quasi-coherent sheaves, coherent sheaves, and vector bundles. The monoidal-category=7parf is moreover expected to be rigid (i.e., every object is dualizable).

 For a morphism    of schemes, there is a-adjunction

 =7 =7 =7=7 

The -functors  and  are supposed to preserve the sub--categories =7qcoh, =7coh, and =7parf, under some finiteness conditions on.

 There exists a notion of short exact sequence in=7, which can be used in order to define a Grothendieck group 



 

=7parf(or more generally a ring spectrum =7parf). This Grothendieck group is called the secondary K-theory of and is expected to possess the usual functori-alities in (at least pull-backs and push-forwards along proper and smooth morphisms).

 There exists a Chern character  

  



for some secondary cohomology group  

. This Chern character is expected to be functorial for pull-backs and to satisfy some version of the Grothendieck–Riemann–Roch formula for push-forwards.

As we will see in Sect. 2, it is not clear how to develop a theory as above, and it seems to us that a theory satisfying all the previous requirements cannot reasonably exist. A crucial remark here is that the situation becomes more easy to handle if the categories A-,[=A ,=A , and6Lare replaced by their derived analogsE, Eqcoh,E

%

coh, andEparf. The categorical sheaf theory we look for should then rather be a derived categorical sheaf theory, and is expected to satisfy the following conditions.

 To any scheme is associated a triangulated--categoryE , of derived categorical sheaves on . Here, by triangulated--category we mean a  -category whose categories of morphisms are endowed with triangulated struc-ture in a way that the composition functors are bi-exacts. The -category E is expected to be a symmetric monoidal-category, in way which is compatible with the triangulated structure. Moreover, we wantE to be a categorification of the derived categoryEof sheaves ofO

--modules on , in the sense that there is a natural triangulated equivalence betweenE and the triangulated category of endomorphisms of the unit object inE .  The -category E  comes equipped with monoidal sub--categories

E qcoh, E coh, and E parf, which are categorifications of the derived categoriesEqcoh,E

%

coh, andEparf, of quasi-coherent com-plexes, bounded coherent sheaves, and perfect complexes. The monoidal -categoryE parfis moreover expected to be rigid (i.e., every object is dualizable).

 For a morphism    of schemes, there is a-adjunction

 E  E  E E  

The -functors  and  are supposed to preserve the sub--categories E qcoh,E coh, andE parf, under some finiteness conditions on.  There exists a notion of short exact sequence inE , which can be used

in order to define a Grothendieck group  

 

E parf(or more generally a ring spectrum E parf). This Grothendieck group is called the secondary K-theory of and is expected to possess the usual functori-alities in (at least pull-backs and push-forwards along proper and smooth morphisms).

 There exists a Chern character  

  



for some secondary cohomology group  

. This Chern character is expected to be functorial for pull-backs and to satisfy some version of the Grothendieck–Riemann–Roch formula for push-forwards.

The purpose of these notes is to give some ideas on how to define such triangulated--categoriesE , the secondary cohomology



, and finally the Chern character. In order to do this,we will follow closely one possible

interpretation of the usual Chern character for vector bundles as being a kind of function on the loop space.

1.3 The Chern Character and the Loop Space

The Chern character we will construct for categorical sheaves is based on the follow-ing interpretation of the usual Chern character. Assume that is a smooth complex algebraic manifold (or more generally a complex algebraic stack) and that6 is a vector bundle on. Let4 



  be a loop in. We do not want to specify want we mean by a loop here, and the notion of loop we will use in the sequel is a rather unconventional one (see3.1). Whatever4truly is, we will think of it as a loop in, at least intuitively. We consider the pull-back4 6, which is a vector bundle on



. Because of the notion of loops we use, this vector bundle is in fact locally constant on



, and thus is completely determined by a monodromy operator( on the fiber6

. The trace of (

is a complex number, and as

4varies inthe loop space of (again the notion of loop space we use is unconventional) we obtain a function= 6on. This function can be seen to be



-equivariant and thus provides an element

= 6O K

 Our claim is that, if the objects



and are defined correctly, then there is a natural identification O K  0v ?@ 

and that= 6is the usual Chern character with values in the algebraic de Rham cohomology of. The conclusion is that= 6can be seen as a



-equivariant function on.

One enlightening example is when is , the quotient stack of a finite group . The our loop space is the quotient stack, for the action of on itself by conjugation. The space of functions on can therefore be identified withC, the space of class functions on. A vector bundle6 on is nothing else than a linear representation of, and the function= 6constructed above is the class function sendingto<B6 6. Therefore, the description of the Chern character above gives back the usual morphismS Csending a linear representation to its class function.

Our construction of the Chern character for a categorical sheaf follows the same ideas. The interesting feature of the above interpretation of the Chern character is that it can be generalized to any setting for which traces of endomorphisms make sense. As we already mentioned,E parfis expected to be a rigid monoidal -category, and thus any endomorphism of an object possesses a trace which is itself an object inEparf  $-. Therefore, if we start with a categorical sheaf on and do the same construction as above, we get a sheaf (rather than a function) on, or more precisely an object inEparf. This sheaf is moreover invariant under the action of



and therefore is an object inE K

 

-equivariant derived category of . This sheaf has itself a Chern character which is an element in

K ?@

, the 

-equivariant de Rham cohomology of. This element is by definition the Chern character of our categorical sheaf. The Chern character should then expected to be a map

=      K ?@ 

1.4 Plan of the Paper

The main purpose of this paper is to make precise all the terms of this construction. For this we will start by the definitions of the-categoriesE , E qcoh, andE parf, but we do not try to defineE cohas the notion of coherence in this categorical setting seems unclear at the moment. The objects inE will be certain sheaves of dg-categories onand our approach to the notion of categorical sheaves heavily relies on the homotopy theory of dg-categories recently studied in [Ta, To2]. In a second part we will recall briefly some ideas of derived algebraic geometry and of derived schemes (and stacks) as introduced in [HAG-II, Lu2]. The loop space of a scheme will then be defined as the derived mapping stack from



Z to. We will argue that the ring of 

-invariant functions on can be naturally identified with

0v =@

, the even de Rham cohomology of, when !has characteristic zero. We will also briefly explain how this can be used in order to interpret the Chern character as we have sketched above. Finally, in a last part we will present the construction of our Chern character for categorical sheaves. One crucial point in this construction is to define an



-equivariant sheaf on the loop space. The construction of the sheaf itself is easy but the fact that it is 

 -equivariant is a delicate question which we leave open in the present work (see4.1 and5.1). Hopefully a detailed proof of the existence of this 



-equivariant sheaf will appear in a future work (see TV1).

2 Categorification of Homological Algebra and dg-Categories

In this section we present our triangulated--categoriesE of derived categor-ical sheaves on some scheme. We will start by an overview of a rather standard way to categorify the theory of modules over some base commutative ring using linear categories. As we will see the notion of-vector spaces appear naturally in this setting as the dualizable objects, exactly in the same way that the dualizable modules are the projective modules of finite rank. After arguing that this notion of -vector space is too rigid a notion to allow for push-forwards, we will consider dg-categories instead and show that they can be used in order to categorify homological algebra in a similar way as linear categories categorify linear algebra. By analogy with the case of modules and linear categories we will consider dualizable objects

as categorified versions of perfect complexes and notice that these are precisely the smooth and proper dg-categories studied in [Ko-So, To-Va]. We will finally define the -categoriesE ,E qcoh, and E parffor a general scheme  by some gluing procedure.

Let!be a commutative base ring. We let A-!be the category of!-modules, considered as a symmetric monoidal category for the tensor product of modules. Recall that an object  A-!is said to be dualizable if the natural morphism

 #

 A(  

is an isomorphism (hereA(denotes the!-module of!-linear morphisms, and

#

 A( !is the dual module). It is easy to see that is dualizable if and only if it is projective and of finite type over!.

A rather standard way to categorify the category A-!is to consider!-linear categories and Morita morphisms. We let=7!be the-category whose objects are small!-linear categories. The category of morphisms between two!-linear cat-egories%and in=7!is defined to be the category of all%



*

-modules (the composition is obtained by the usual tensor product of bi-modules). The tensor product of linear categories endow=7!with a structure of a symmetric monoidal -category for which!, the!-linear category freely generated by one object, is the unit. We have

$-Q#

! A-!, showing that=7!is a categorification of A-!. To obtain a categorification of

A- 

!, the category of projective !-modules of finite type, we consider the sub--category of=7!with the same objects but for which the category of morphisms from%to is the full sub-category of the category of%



*

-modules whose objects are bi-modules such that for any7  %the

*

-module 7is projective of finite type (i.e., a retract of a finite sum of representable

*

-modules). We let=7 

! =7!be this sub--category, which is again a symmetric monoidal-category for tensor product of linear categories. By definition we have

$-Q#   ! A-  !. However, the tensor category

A- 

!is a rigid tensor category in the sense that every object is dualizable, but not every object in=7



!is dualizable. We therefore consider=7

#

!the full sub--category of dualizable objects in=7 

!. Then, =7

#

!is a rigid monoidal-category which is a categorification of A- 

!. It can be checked that a linear category%is in=7

#

!if and only if it is equiv-alent in =7! to an associative !-algebra (as usual considered as a linear category with a unique object) satisfying the following two conditions.

1. The!-module is projective and of finite type over!. 2. For any associative!-algebra%, a 



%-module is projective of finite type if and only if it is so as a%-module.

These conditions are also equivalent to the following two conditions. 1. The!-module is projective and of finite type over!.

2. The  

*

-module is projective. When!is a field, an object in=7

#

!is nothing else than a finite dimensional !-algebra which is universally semi-simple (i.e., such that 

 !



is semi-simple for any field extension! !



). In general, an object in=7 #

!is a flat family of universally semi-simple finite dimensional algebras over,L!. In particular, if!is an algebraically closed field any object in=7

#

!is equivalent to! 

, or in other words is a-vector space of finite dimension in the sense of Kapranov– Voevodsky (see for instance [Ba-Du-Ro]). For a general commutative ring!, the -category=7

#

!is a reasonable generalization of the notion of-vector spaces and can be called the-category of-vector bundles on,L!.

One major problem with this notion of-vector bundles is the lack of push-forwards in general. For instance, let be a smooth and proper algebraic variety over some algebraically closed field!. We can consider6L, the trivial-vector bundle of rankover, which is the stack in categories sending a Zariski open  to the linear category6Lof vector bundles over. The push-forward of this trivial-vector bundle along the structure morphism  ,L!is the !-linear category of global sections of6L, or in other words the!-linear category 6Lof vector bundles on. This is an object in=7!, but is definitely not in=7

#

!. The linear category6Lis big enough to convince anyone that it cannot be finite dimensional in any reasonable sense. This shows that the global sections of a-vector bundle on a smooth and proper variety is in general not a -vector bundle over the base field, and that in general it is hopeless to expect a good theory of proper push-forwards in this setting.

A major observation in this work is that considering a categorification ofE! instead of A-!, which is what we call a categorification of homological alge-bra, solves the problem mentioned above concerning push-forwards. Recall that a dg-category (over some base commutative ring !) is a category enriched over the category of complexes of!-modules (see [Ta]). For a dg-category< we can define its category of <-dg-modules as well as its derived category E< by formally inverting quasi-isomorphisms between dg-modules (see [To2]). For two dg-categories<

and <

we can form their tensor product <    < , as well as their derived tensor product<

 

L

 <

(see [To2]). We now define a

-catgeoryE ! whose objects are dg-categories and whose category of morphisms from<

to < is E<   L  < *

. The composition of morphisms is defined using the derived tensor product  L Y  E<   L  < * E<  L  < *   E<   L  < *  

Finally, the derived tensor product of dg-categories endowsE !with a structure of a symmetric monoidal-category.

The symmetric monoidal-categoryE !is a categorification of the derived categoryE!as we have by definition

To obtain a categorification ofEparf!, the perfect derived category, we consider the sub--categoryE 



!having the same of objects asE !itself but for which the category of morphisms from<

to <

in E 



!is the full sub-category ofE<   L  < *

of bi-dg-modules/ such that for all  <

 the object

/  E< *

is compact (in the sense of triangulated categories, see [Ne]). The symmetric monoidal structure onE !restricts to a symmetric monoidal structure onE 

 !, and we have $-?    Eparf!

as an object ofE!is compact if and only if it is a perfect complex. Finally, the symmetric monoidal-categoryE 



!is not rigid and we thus considerE  #

!, the full sub--category consisting of rigid objects in E 



!. By construction, E 

#

!is a rigid symmetric monoidal-category and we have

$-?  !  Eparf! The -categoryE  #

!will be our categorification of Eparf! and its objects should be thought as perfect derived categorical sheaves on the scheme,L!.

It is possible to show that an object< ofE  

!belongs to E  #

!if and only if it is equivalent to an associative dg-algebra , considered as usual as a dg-category with a unique object, satisfying the following two conditions.

1. The underlying complex of!-modules of is perfect. 2. The object E 

L



*

is compact. In other words, a dg-category<belongs toE 

#

!if and only it is Morita equi-valent to a smooth (conditionabove) and proper (conditionabove) dg-algebra . Such dg-categories are also often called saturated (see [Ko-So, To-Va]).

AsE  #

!is a rigid symmetric monoidal-category we can define, for any object< a trace morphism

<B $-?  !  <  $-?  !  Eparf! Now, the category

$-?  !



<can be naturally identified withE< 

L  < *  , the full sub-category ofE< 

L

 <

*

of compact objects (here we use that< is saturated and the results of [To-Va, Sect. 2.2]). The trace morphism is then a functor

<B E<  L  < *    Eparf!

which can be seen to be isomorphic to the functor sending a bi-dg-module to its Hochschild complex< Eparf!. In particular, the rank of an object <  E 

#

!, which by definition is the trace of its identity, is its Hochschild complex<Eparf!.

To finish this section we present the global versions of the-categoriesE ! andE 

#

!over some base scheme. We letC7B%be the small site of affine Zariski open sub-schemes of. We start to define a category-7 consisting of the following data

1. For any,L%  inC7B%, a dg-category< > over

%. 2. For any,L 6  ,L%    morphism in C7B%a

morphism of dg-categories over% B > T < >  < T 

These data should moreover satisfy the equationB T]

ÆB > T

B

> ] for any inclusion of affine opens? 6  . The morphisms in-7are defined in an obvious way as families of dg-functors commuting with theB

> T’s.

For<  -7we define a category A-<of<-dg-modules in the following way. Its objects consist of the following data

1. For any,L%  inC7B%, a<

>-dg-module

>.

2. For any,L 6  ,L%    morphism in C7B%a morphism of< >-dg-modules ( > T  >  B > T  T 

These data should moreover satisfy the usual cocycle equation forB > T ( T] Æ ( > T ( > ]. Morphisms in

A-<are simply defined as families of morphisms of dg-modules commuting with the(

> T’s. Such a morphism

   

in A-<is a quasi-isomorphism if it is a stalkwise quasi-isomorphism (note that and



are complexes of presheaves ofO--modules). We denote by

E<the category obtained from A-<by formally inverting these quasi-isomorphisms.

We now define a-categoryE whose objects are the objects of- 7, and whose category of morphisms from<

to < is E<   L O < * (we pass on the technical point of defining this derived tensor product overO-, one possibility being to endow-7with a model category structure and to use a cofi-brant replacement). The compositions of morphisms inE is given by the usual derived tensor product. The derived tensor product endowsE with a structure of a symmetric monoidal-category and we have by construction

 $-? -

E

whereEis the (unbounded) derived category of allO

--modules on .

Definition 2.1.1. An object< E is quasi-coherent if for any inclusion of affine open subschemes

,L 6 ,L%  the induced morphism

< >  2  < T is a Morita equivalence of dg-categories.

2. A morphism<   < , corresponding to an object  E<   L O < * , is called quasi-coherent if its underlying complex of O--modules is with quasi-coherent cohomology sheaves.

The sub--category ofE consisting of coherent objects and quasi-coherent morphisms is denoted byE qcoh. It is called the -category of quasi-coherent derived categorical sheaves on.

Let< and

<

be two objects in

E qcoh. We consider the full sub-category of E<   L O < *

consisting of objects such that for any Zariski open,L%   and any object <

 

>, the induced dg-module

7 E< *  >  is compact. This defines a sub--category ofE qcoh, denoted byE 



qcohand will be called the sub--category of compact morphisms. The symmetric monoidal structure onE restricts to a symmetric monoidal structure onE qcohand onE 



qcoh.

Definition 2.2. The-category of perfect derived categorical sheaves is the full sub--catgeory of E 



qcoh consisting of dualizable objects. It is denoted by E parf.

By construction,E parfis a symmetric monoidal-category with

$-? parf-

Eparf

It is possible to show that an object<  Eqcohbelongs toE parfif and only if for any affine Zariski open subscheme,L%  , the dg-category <

> is saturated (i.e., belongs to E 

# %).

For a morphism of schemes    it is possible to define a-adjunction  E  E  E E  

Moreover, preserves quasi-coherent objects, quasi-coherent morphisms, as well as the sub--categories of compact morphisms and perfect objects. When the mor-phism is quasi-compact and quasi-separated, we think that it is possible to prove that preserves quasi-coherent objects and quasi-coherent morphisms as well as the sub--category of compact morphisms. We also guess that will preserve perfect objects when is smooth and proper, but this would require a precise inves-tigation. As a typical example, the direct image of the unit  E parfby is the presheaf of dg-categories sending,L%    to the dg-category parf

X

of perfect complexes over the scheme 

  X

. When  is smooth and proper it is known that the dg-categoryparf

X

is in fact saturated (see [To2, Sect. 8.3]). This shows that is a perfect derived sheaf on  and provides an evidence that preserves perfect object. These functoriality statements will be considered in more details in a future work.

3 Loop Spaces in Derived Algebraic Geometry

In this section we present a version of the loop space of a scheme (or more gener-ally of an algebraic stack) based on derived algebraic geometry. For us the circle 



is defined to be the quotient stack Z, where Z is considered as a constant sheaf of groups. For any scheme, the mapping stack Map



is then equiv-alent to , as the coarse moduli space of



is simply a point. In other words, with this definition of the circle there are no interesting loops on a scheme . However, we will explain in the sequel that there exists an interesting derived mapping stackRMap



, which is now a derived scheme and which is non-trivial. This derived mapping stack will be our loop space. In this section we recall briefly the notions from derived algebraic geometry needed in order to define the objectRMap



. We will also explain the relation between the cohomology of RMap



and cyclic homology of the scheme. Let! be a base commutative ring and denote by Sch

, resp. St, the category of schemes over ! and the model category of stacks over ! ( [HAG-II, 2.1.1] or [To1, Sect. 2, 3]), for the ´etale topology. We recall that the homotopy category HoSt



contains as full sub-categories the category of sheaves of sets on Sch  as well as the-truncation of the-category of stacks in groupoids (in the sense of [La-Mo]). In particular, the homotopy category of stacks HoSt



contains the category of schemes and of Artin stacks as full sub-categories. In what follows we will always consider these two categories as embedded in HoSt



. Finally, recall that the category HoSt



possesses internal Hom’s, that will be denoted by Map. As explained in [HAG-II, Chap. 2.2] (see also [To1, Sect. 4] for an overview), there is also a model category dSt of derived stacks over

! for the strong ´etale topology. The derived affine objects are simplicial!-algebras, and the model cat-egory of simplicial !-algebras will be denoted by salg

. The opposite (model) category is denoted by dAff. Derived stacks can be identified with objects in the homotopy category HodSt



which in turn can be identified with the full subcate-gory of the homotopy catesubcate-gory of simplicial presheaves on dAffwhose objects are weak equivalences’ preserving simplicial presheaves/ having strong ´etale descent i.e., such that, for any ´etale homotopy hypercover



 in dAff

 ( [HAG-I, Definition 3.2.3, 4.4.1]), the canonical map

/ holim/ 



is a weak equivalence of simplicial sets. The derived Yoneda functor induces a fully faithful functor on homotopy categories

RSpecHodAff 

" HodSt 

 % RSpec%   Mapsalg 

% 

where Mapsalg



denotes the mapping spaces of the model category salg (there-fore Mapsalg



% A([% , whereA(denotes the natural simplicial Hom’s of salg and

[% is a cofibrant model for %). Those derived stacks belonging to the essential image ofRSpec will be called affine derived stacks.

The category HodSt 

of derived stacks has a lot of important properties. First of all, being the homotopy category of a model category, it has derived colimits and limits (denoted as hocolim and holim). In particular, given any pair of maps / and between derived stacks, there is a derived fiber product stack

holim/    ' / 

 K

. As our base ring! is not assumed to be a field, the direct product in the model category dStis not exact and should also be derived. The derived direct product of two derived stacks/ andwill be denoted by/



. This derived product is the categorical product in the homotopy category HodSt



. The category HodSt 

also admits internal Hom’s, i.e., for any pair of derived stacks/ andthere is a derived mapping stack denoted as

RMap/ with the property that

/RMap /  



functorially in/,, and.

The inclusion functor + of commutative !-algebras into salg

 (as constant simplicial algebras) induces a pair'



of (left,right) adjoint functors

  + HodSt   HoSt   'L+ ` HoSt   HodSt  

It can be proved that' is fully faithful. In particular we can, and will, view any stack as a derived stack (we will most of the time omit to mention the functor' and consider HoSt



 as embedded in HodSt 

). The truncation functor   acts on affine derived stacks as 



RSpec% Spec 

%. It is important to note that the inclusion functor' does not preserve derived internal hom’s nor derived fibered products. This is a crucial point in derived algebraic geometry: derived tan-gent spaces and derived fiber products of usual schemes or stacks are really derived objects. The derived tangent space of an Artin stack viewed as a derived stack via 'is the dual of its cotangent complex while the derived fiber product of, say, two affine schemes viewed as two derived stacks is given by the derived tensor product of the corresponding commutative algebras

'Spec  Spec@  'Spec<RSpec L @ <

Both for stacks and derived stacks there is a notion of being geometric ([HAGII, 1.3, 2.2.3]), depending, among other things, on the choice of a notion of smooth morphism between the affine pieces. For morphisms of commutative!-algebras this is the usual notion of smooth morphism, while in the derived case, a mor-phism % of simplicial !-algebras is said to be strongly smooth if the induced map 



% 



is a smooth morphism of commutative rings, and  %

42 



  . The notion of geometric stack is strictly related to the notion of Artin stack ([HAG-II, Proposition 2.1.2.1]). Any geometric derived stack

has a cotangent complex ( [HAG-II, Corollary 2.2.3.3]). Moreover, both functors  and'preserve geometricity.

Let BZ be the classifying stack of the constant group scheme Z. We view BZ as an object of St, i.e., as the stack associated to the constant simplicial presheaf

BZalg 

SSetsS BZ

where by abuse of notations, we have also denoted as BZ the classifying simplicial set, i.e., the nerve of the (discrete) groupZ. Such a nerve is naturally a pointed simplicial set, and we callthat point.

Definition 3.1. Letbe a derived stack over!. The derived loop stack ofis the derived stack

LRMapBZ

We will be mostly interested in the case where is a scheme or an algebraic (underived) stack. Taking into account the homotopy equivalence

BZ      

we see that we have

L   - # - where maps to  

 diagonally and the homotopy fiber product is taken in

dSt. Evaluation at

BZ yields a canonical map of derived stacks

, L  

On the other hand, since the limit maps canonically to the homotopy limit, we get a canonical morphism of derived stacks L, a section of,, describing as the “constant loops” in L.

If is an affine scheme over!, Spec%with%a commutative!-algebra, we get that

L RSpec%

L

2L2 %

where the derived tensor product is taken in the model category salg. One way to rephrase this is by saying that “functions” on LSpec%are Hochschild homology classes of%with values in%itself. Precisely, we have

OLRA(LA 

HH%%

where HH%% is the simplicial set obtained from the complex of Hochschild homology of%by the Dold–Kan correspondence, andRA(denotes the natural enrichment of HodSt



into HoL. When is a general scheme thenOL can be identified with the Hochschild homology complex of, and we have

 

OL 



In particular, when is a smooth and!is of characteristic zero, the Hochschild– Kostant–Rosenberg theorem implies that

  OL.    Z  -I  The stack 

BZ is a group stack, and it acts naturally on L for any derived stack by “rotating the loops.” More precisely, there is a model category dSt

K I, or



-equivariant stacks, and L is naturally an object in the homotopy category HodSt

K I

. This way, the simplicial algebra of functionsOLis naturally an 



-equivariant simplicial algebra, and thus can also be considered as an  

-equivariant complex or in other words as an object inE

K

!, the 

-equivariant derived category of!. The categoryE

K

!is also naturally equivalent toE!F, the derived category of the dg-algebra!F freely generated by an element F of degreeand withF

. The derived categoryE K

!is thus naturally equiv-alent to the derived category of mixed complexes (see [Lo]) (multiplication byF providing the second differential). When!is of characteristic zero and Spec% one can show (see TV2) that

  OL K  0v =@ % where K

denotes the simplicial set of homotopy fixed points of an 

-equivariant simplicial set . In other words, there is a natural identification between 

 -invariant functions on L and even de Rham cohomology of . This statement of course can be generalized to the case of a scheme.

Proposition 3.2. For a scheme we have

  OL K  0v =@ 

In the first version of this paper, we conjectured a similar statement, for! of any characteristic and with even de Rham cohomology replaced by negative cyclic homology. At the moment, however, we are only able to prove the weaker results above (see TV2).

The even part of de Rham cohomology ofcan be identified with 

-equivariant functions on the derived loop space L. This fact can also be generalized to the case where is a smooth Deligne–Mumford stack over!(again assumed to be of characteristic zero), but the right hand side should rather be replaced by the (even part of) de Rham orbifold cohomology of, which is the de Rham cohomology of the inertia stackP 

 L.

To finish this part, we would like to mention that the construction of the Chern character for vector bundles we suggested in Sect. 1 can now be made precise, and through the identification of Proposition3.2, this Chern character coincides with

the usual one. We start with a vector bundle6 on  and we consider its pull-back , 6on L, which is a vector bundle on the derived scheme L. This vector bundle, 6comes naturally equipped with an automorphism u. This fol-lows by considering the evaluation morphism 



L  , and the vector bundle 6. As 



Z, a vector bundle on 

L consists precisely of a vector bundle on L together with an action ofZ, or in other words together with an automorphism. We can then consider the trace of u, which is an element in 



OL  

. A difficult issue here is to argue that this function<Bu has a natural refinement to an

 -invariant function<Bu    OL K    0v =@

!, which is the Chern character of6. The 

-invariance of<Buwill be studied in a future work, and we refer to our last section below, for some comments about how this would follow from the general theory of rigid tensor-categories.

4 Construction of the Chern Character

We are now ready to sketch the construction of our Chern character for a derived categorical sheaf. This construction simply follows the lines we have just sketched for vector bundles. We will meet the same difficult issue of the existence of an 



-invariant refinement of the trace, and we will leave this question as an conjecture. However, in the next section we will explain how this would follow from a very general fact about rigid monoidal-categories.

Let<  E parfbe a perfect derived categorical sheaf on some scheme (or more generally on somealgebraicc stack). We consider the natural morphism ,L  and we consider, <, which is a perfect derived categorical sheaf on L. We have not defined the notions of categorical sheaves on derived schemes or derived stacks but this is rather straightforward. As in the case of vector bundles explained in the last section, the object, <comes naturally equipped with an autoequivalence u. This again follows from the fact that a derived categorical sheaf on



Lis the same thing as a derived categorical sheaf on Ltogether with an autoequivalence. We consider the trace of u in order to get a perfect complex on the derived loop space

<Bu

$-? parfL-

EparfL

The main technical difficulty here is to show that<Bupossesses a natural lift as an



-equivariant complex on L. We leave this as a conjecture. Conjecture 4.1. The complex<Buhas a natural lift

<B K uE K parfL whereE K parfLis the 

The above conjecture is not very precise as the claim is not that a lift simply exists, but rather than there exists a natural one. One of the difficulty in the conjec-ture above is that it seems difficult to characterize the required lift by some specific properties. We will see however that the conjecture can be reduced to a general conjecture about rigid monoidal-categories.

Assuming Conjecture4.1, we have<B K

uand we now consider its class in the Grothendieck group of the triangulated categoryE

K

parfL. This is our definition of the categorical Chern character of<.

Definition 4.2. The categorical Chern character of< is

= # <<B K u K  L  E K parfL The categorical Chern character=

#

<can be itself refined into a cohomo-logical Chern character by using now the



-equivariant Chern character map for 



equivariantt perfect complexes on L. We skip some technical details here but the final result is an element

= * <= K = # <  OL   K K   where L    RMap   

is now the derived double loop space of. The space  OL   K K 

can reasonably be called the secondary negative cyclic homology of  and should be thought (and actually is) the



-equivariant negative cyclic homology of L. We therefore have

= * <= K  L

Definition 4.3. The cohomological Chern character of< is

= * <= K = # <= K  L  OL   K K  defined above.

Obviously, it is furthermore expected that the constructions<  = #

< and<  =

*

<satisfy standard properties such as additivity, multiplicativity and functoriality with respect to pull-backs. The most general version of our Chern character map should be a morphism of commutative ring spectra

= #

  K

L

where is a ring spectrum constructed using a certain Waldhausen category of perfect derived categorical sheaves onand

K

Lis the -theory spectrum of



-equivariant perfect complexes on L. This aspect of the Chern character will be investigated in more details in a future work.

5 Final Comments

On 

-equivariant trace maps. Our Conjecture4.1can be clarified using the lan-guage of higher categories. Recall that a-category is an-category in which all$-morphisms are invertible (up to higher morphisms) as soon as$ 5 . There exist several well behaved models for the theory of-categories, such as sim-plicially enriched categories, quasi-categories, Segal categories, and Rezk’s spaces. We refer to [Ber1] for an overview of these various notions. What we will say below can be done in any of these theories, but, to fix ideas, we will work withS-categories (i.e., simplicially enriched categories).

We will be usingAS=7the homotopy category ofS-categories, which is the category obtained from the category ofS-categories and S-functors by inverting the (Dwyer–Kan) equivalences (a mixture between weak equivalences of simplicial sets and categorical equivalences). An important property ofAS =7is that it is cartesian closed (see [To2] for the corresponding statement for dg-categories whose proof is similar). In particular, for twoS-categories= and=



we can construct an S-category RA(==



with the property that

=  RA(==  =  ==  

wheredenote the Hom sets ofAS=7. AnyS-category= gives rise to a genuine category=with the same objects and whose sets of morphisms are the connected components of the simplicial sets of morphisms of=.

We let: be the category of pointed finite sets and pointed maps. The finite set$ pointed at  will be denoted by$

. Now, a symmetric monoidal S-category is a functor

:  S=7 such that for any$the so-called Segal morphism

$     

induced by the various projections$



sending'  $ to and everything else to, is an equivalence ofS-categories. The full sub-category of the homotopy category of functorsAS=7

C

consisting of symmetric monoidal S-categories will be denoted byAS=7



. As the categoryAS=7is a model for the homotopy category of-categories, the categoryAS=7

  is a model for the homotopy category of symmetric monoidal-categories. For AS=7



we will again use to denote its underlyingS-category 

. TheS-category 

has a natural structure of a commutative monoid in AS=7. This monoid structure will be denoted by.

We say that a symmetric monoidalS-category is rigid if for any object there is anobjectc

#

 and a morphism  #

such that for any pair of objects.z , the induced morphism of simplicial sets

.z . # z #  .z # 

is an equivalence. In particular, the identity of #

provides a trace morphism 

#

(. #

, z). Therefore, for any rigid symmetric monoidalS-category and an object we can define a trace morphism

<B 

   #

 

Let be a fixed rigid symmetric monoidal S-category and  

Z be the groupoid with a unique object with Z as automorphism group. The category

 is an abelian group object in categories and therefore can be considered as a group object inS-categories. The S-category of functors RA(



 is denoted by 



, and is equipped with a natural action of 

. We consider the sub-S-category of invertible (up to homotopy) morphisms in 



whose classifying space is an 



-equivariant simplicial set. We denote this simplicial set by (“” stands for “loops”). It is possible to collect all the trace morphisms<B

defined above into a unique morphism of simplicial sets (well defined inAL)

<B  

Note that the connected components of are in one to one correspondence with the set of equivalences classes of pairsu, consisting of an objectin and an autoequivalence u of. The morphism<B is such that<Bu <B

 u  



 . We are in fact convinced that the trace map<Bcan be made equivari-ant for the action of



on , functorially in . To make a precise conjecture, we considerS=7

)

the category of all rigid symmetric monoidalS-categories (note thatS=7

)

is not a homotopy category, it is simply a full sub-category of S=7

C

). We have two functors S=7 )    L to the category of 

-equivariant simplicial sets. The first one sends to together with its natural action of



. The second one sends to with the trivial



-action. These two functors are considered as objects inA/u$S =7

) 



L, the homotopy category of functors. Let us denote these two objects by   and  .

Conjecture 5.1. There exists a morphism inA/u$S=7 )

 

L

<B 

in such a way that for any rigid symmetric monoidalS-category< the induced morphism of simplicial sets

<B  

is the trace map described above.

It can be shown that Conjecture5.1implies Conjecture4.1. In fact the tensor -categoriesE parfare the -truncation of natural rigid symmetric monoidal -categories, which can also be considered as-categories by only con-sidering invertible higher morphisms. An application of the above conjecture to these rigid symmetric monoidal-categories give a solution to Conjecture

4.1, but this will be explained in more detailed in a future work. To finish this part on rigid -categories let us mention that a recent work of Lurie and Hopkins on universal properties of-categories of-bordisms seems to solve Conjecture5.1( [Lu3]). We think we have another solution to the part of Conjec-ture5.1concerned with the rigid symmetric monoidal-category of saturated dg-categories, which is surely enough to imply Conjecture4.1. This again will be explained in a future work.

The Chern character as part of a 1-TFT over. The recent work of Lurie and Hopkins (see [Lu3]) mentioned above also allows us to fit our Chern character into the framework of 1-dimensional topological field theories. Indeed, let  be a scheme and<  E parfbe a perfect derived categorical sheaf on. It is represented by a morphism of derived stacks

<   E parf

where E parf is considered here as a symmetric monoidal -category as explained above. The object E parf is thus a derived stack in rigid symmetric monoidal-categories, and the results of [Lu3] imply that the morphism< induces a well defined morphism of derived stacks in rigid symmetric monoidal -categories

 AB- E 

parf

Here AB-is the categorical object in derived stacks of relative-bordisms over (see [Lu3] for details). In terms of functor of points it is defined by sending a simplicial commutative algebra%to the -category AB-%of relative-bordisms over the space%. It is not hard to see that AB- is itself a derived algebraic stack, and thus a rigid symmetric monoidal categorical object in algebraic derived stacks. The morphism

 AB- E parf

is therefore a relative 1-dimensional TQFT over and takes its values inE parf. Our Chern character can be easily reconstructed from this-dimensional field the-ory by considering its restriction to the space of endomorphisms of the unit object # . Indeed, this space contains as a component the moduli space of oriented cir-cles over, which is nothing else than



(the quotient stack of by the action of



of derived stacks    $-? parf Eparf whose datum is equivalent to the datum of an object inE

K

parf. This object turns out to be isomorphic to our=

# <.

Relations with variations of Hodge structures (see TV1). The derived loop space Land the



-equivariant derived categoryE K

parfLhave already been studied in [Be-Na]. In this work the categoryE

K

parfLis identified with a certain derived cat-egory of modules over the Rees algebra of differential operators on (when, say,  is smooth over! of characteristic zero). We do not claim to fully understand this identification but it seems clear that objects inE

K

parfLcan be identified with some kind of filtered complexes ofE-modules on. Indeed, when ,L%is smooth and affine over!of characteristic zero, the categoryE

K

parfLcan be iden-tified with the derived category of dg-modules over the dg-algebraZ

2

>, generated by the graded algebra of differential forms (hereZ



2sits in degree

') together with an extra element>of degree acting as the de Rham differential onZ

2. This dg-algebra is itself Kozsul dual to the Rees algebraR- of differential operators ondefined as follows. The dg-algebraR

- is generated by . D  -'together with an element u of degreeacting by the natural inclusionsD

 -D   -(here D 

-denotes as usual the ring of differential operators of degree less than '). We thus have an equivalenceE

K

parf  EparfR

-. Now, apart from its unusual grading,R- is essentially the Rees algebra associated to the filtered ringD-, and thusEparfR

-is essentially the derived category of filteredE-modules (which are perfect over, and with this unusual grading).

Using this identification our categorical Chern character = #

< probably encodes the data of the negative cyclic complex=



<of< over together with its Gauss–Manin connection and Hodge filtration. In other words,=

# < seems to be nothing more than the variation of Hodge structures induced by the family of dg-categories< over . As far as we know the construction of such a structure of variations of Hodge structures on the family of complexes of cyclic homology associated to a family of saturated dg-categories is a new result (see how-ever [Ge] for the construction of a Gauss–Manin connection on cyclic homology). We also think it is a remarkably nice fact that variations of Hodge structures appear naturally out of the construction of our Chern character for categorical sheaves.

It is certainly possible to describe the cohomological Chern character of4.3 using this point of view of Hodge structure. Indeed,=

K 

Lis close to be the 



-equivariant de Rham cohomology of L, and using a localization formula it is probably possible to relate=

K  Lwith    , where  is periodic cyclic homology of and is a formalparameterr. We expect at least a morphism = K  L      The image of= *

<under this map should then be closely related to the Hodge polynomial of<, that is  = B  =  <  , whereB  =  <is the,-th

graded piece of the Hodge filtration of Hochschild homology and= is the usual Chern character for sheaves on.

Back to elliptic cohomology ? In the Introduction we mentioned that our motiva-tion for thinking about categorical sheaf theory originated from elliptic cohomology. However, our choice to work in the context of algebraic geometry drove us rather far from elliptic cohomology and it is at the moment unclear whether our work on the Chern character can really bring any new insight on elliptic cohomology. About this we would like to make the following remark. Since what we have been consid-ering are categorified version of algebraic vector bundles, it seems rather clear that what we have done so far might have some relations with what could be called alge-braic elliptic cohomology (by analogy with the distinction between algealge-braic and topological -theory). However, the work of Walker shows that algebraic -theory determines completely topological -theory (see [Wa]), and that it is possible to recover topological -theory from algebraic -theory by an explicit construction. Such a striking fact suggests the possibility that understanding enough about the algebraic version of elliptic cohomology could also provide some new insights on usual elliptic cohomology. We leave this vague idea to future investigations.

In a similar vein, as observed by the referee, the analogy with the chromatic picture, hinted at in the Introduction, would suggest the existence of some kind of action of the second Morava stabilizer groupG on the ring spectrum

 of Sect. 4 (suitably ,-completed at some prime,), and also the possibility that =

#

  K

Lcould detect some kind of v

-periodicity (i.e., some evidence of chromatic type) in

K

L. Unfortunately, at the moment, we are not able to state these suggestions more precisely, let alone giving an answer.

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